Research

My PhD thesis involved the representation theory of symmetric groups over fields of positive characteristic. The so called Specht modules play a central role in the theory, as they form a complete list of non-isomorphic simple modules over the complex numbers and many modules over positive characteristic have filtrations by Specht modules. Researchers in this field try to understand Specht modules to better understand the representation theory.

Papers

M. Rosas. On the structure of Specht modules in the principal block of FS_3p. Journal of Algebra 448 (2016) pp. 446-447.

https://doi.org/10.1016/j.jalgebra.2015.10.003

(You can read the unpublished version below.)

Current projects

• I am currently working on a project with three other collaborators which involves studying the branching (induction and restriction) of Lie super algebras to special Lie super subalgebras.
• Study the decomposition numbers of simple modules which lie in weight 3 blocks of symmetric group algebras over fields of positive characteristic. Determining the decomposition numbers of the simple modules is widely regarded as the most important open problem in the representation theory of symmetric groups.

Future projects

• Linear Algebra Pedagogy: Work with other departments to develop best teaching practices for linear algebra courses. I've taught introductory linear algebra a number of times, and students tend to struggle when trying to apply what they have learned. By apply, I mean apply the linear algebra concepts to modeling problems, not just test well. I would very much like to see more attention given to linear algebra pedagogy.
• Repository of problems: I would like to work alongside other departments in creating a repository of problems for different majors or interests. For example, if an economics major is taking linear algebra the instructor can offer the student examples from the "economics" problems developed to serve the interest of the student. Moreover, other departments can tell the math department what they'd like their majors see in a service course.